Characters of the Norm-One Units of Local Division Algebras of Prime Degree

Abstract

We give an explicit construction of all complex continuous irreducible characters of the group ${\mathrm{SL}}_1(D)$, where $D$ is a division algebra of prime degree $\ell$ over a local field of odd residual characteristic different than $\ell$. For $\ell$ odd, we show that all such characters of ${\mathrm{SL}}_1(D)$ are induced from linear characters of compact-open subgroups of ${\mathrm{SL}}_1(D)$. We also compute an explicit formula for the representation zeta function of ${\mathrm{SL}}_1(D)$.

33 pages; Methods and organization section added andseveral minor changes made; final version

References in corpus (2)

• Representation zeta functions of compact p-adic analytic groups and arithmetic groups
• Representation zeta functions of some compact p-adic analytic groups